Above the Noise Floor

Here is the relation, unchanged. Two thermal nodes, the core temperature eliminated symbolically, leaving something testable from the case sensor and the known heat input alone:

Now put a real sensor on it. A thermocouple with a quarter of a degree of noise, sampled every two seconds. The relation contains a second derivative of that signal, and a second derivative of noise is a catastrophe.

That is not a tuning problem. No threshold rescues a statistic whose noise is four hundred times its signal. If you build this monitor and put it on a vehicle it will alarm constantly and detect nothing, and the failure will look like a software defect rather than what it is, which is a design error made three steps earlier.

The pleasant part is that the size of the disaster is predictable in closed form, so you can see it coming before writing any code.

A residual is a linear operator applied to the measurement. Independent noise of standard deviation sigma at each sample, passed through a discrete second derivative built from two central differences, comes out with variance six over sixteen h to the fourth. The relation multiplies that by the product of the two capacitances:

Three things follow immediately, and all three are visible in the formula rather than discovered by experiment.

• The fourth power of the sample interval. Halving the sample period multiplies the noise by sixteen. Sampling faster makes this monitor worse, which is the opposite of everyone's instinct.
• The capacitance product is a pure gain. A large thermal mass, which is physically desirable, is a liability for this particular statistic. Nothing about the physics changed; the way the relation was normalized did.
• The order of the relation is the whole problem. A relation of order n in the measured signal carries a noise gain going as h to the minus n. This is Section 12 of the parent page, the finite difference trap, arriving where it does the most damage.

Here is the part that changes how you should think about all of this. The catastrophe of Section 1 is a property of the statistic, not of the measurement. The fault information was never destroyed. It was discarded.

Watch what happens when you write it out. Let the fault change the measured signal by some waveform. The measurement is that plus noise, and the residual is the relation applied to both:

The fault and the noise pass through the same operator, so in the ratio that governs optimal detection the operator divides out. The best achievable detection performance depends only on how much the fault changes the measured signal, measured against the sensor noise, and not at all on how you chose to write the relation.

This is worth stating plainly because it reframes the whole exercise. Deriving the relation is a modeling achievement. Turning it into a detector is a separate design problem, and the second one is where monitors are usually lost. The symbolic work of the parent page tells you what is knowable. It does not tell you how to know it, and assuming that a threshold on the derived expression is good enough is the mistake this page exists to name.

Same model, same data, same noise. Four different things to compute from it, scored identically by how well they separate healthy from faulted.

• Peak of the raw relation. The obvious thing. Separation of essentially zero. Detects nothing above chance.
• Peak of a low-pass filtered relation. The obvious fix, and it barely helps. Filtering removes noise and the fault signal together, because a relation dominated by a second derivative puts both in the same band. Tuning the corner frequency across two decades never gets the separation above 0.02.
• Matched filter. Correlate the residual against the residual waveform the fault would produce. Separation between 220 and 3,800 depending on the fault, and perfect detection at any sensible false alarm rate. This is the ceiling from Section 3, reached.
• Parameter estimation. Fit the suspect parameter and test whether it moved. Equivalent to the matched filter when the fault size is unknown, which is the usual case, and it returns a severity rather than a yes or no.

There is a loose end in the parent page that this explains. Section 41 there ran an extended Kalman filter on this same model, with this same quarter degree of sensor noise, and it tracked a drifting conductance to two thousandths of a watt per kelvin. Meanwhile the residual of Section 36 could not see a thirty percent fault.

Both are true, and they are the same story from two ends. The Kalman filter is, for a linear Gaussian problem, the optimal estimator. It is quietly doing the matched filtering of Section 3: weighting each measurement by its information content, integrating over the whole history, and never differentiating anything. The residual was throwing that away one sample at a time.

The estimator route costs more computation and needs a covariance you can defend. What it buys is that it uses the information rather than sampling it, and on any model where the fault is slow relative to the observation window that difference is not a refinement. It is the difference between working and not working.

Thresholds usually get set by running the monitor on some healthy data, seeing the biggest excursion, and adding margin. That produces a number nobody can defend and that nobody can adjust rationally when the fleet grows.

The defensible way round is to start from what you are willing to spend on false alarms and derive the threshold from the noise model you already have. A monitor testing once an hour for four thousand hours makes four thousand opportunities to be wrong. If the budget is one percent of units false alarming over that life, the per-test rate has to be about two and a half parts in a million, which is four and a half sigma, not three.

Three sigma, the number everyone reaches for, gives a false alarm on essentially every unit over that many tests. The parent page's own simulation confirmed it: ninety-nine runs in a hundred tripped.

Every wear mechanism produces a slow ramp, and a fixed threshold is a poor detector of slow ramps. It looks at each sample alone and throws away the fact that the last four hundred samples were all slightly high in the same direction.

The cumulative sum test keeps that memory. It accumulates the residual minus a small slack term, floors at zero, and alarms when the accumulation crosses a limit. Tuned to the same false alarm budget as the fixed threshold, it detects the same drift considerably earlier, and the advantage grows the slower the drift is.

Read the last row. On the slowest drift, which is the one a real component actually exhibits, the cumulative test finds the fault three times sooner and at a third of the severity, for no extra sensor and no extra false alarms. The only cost is one stored accumulator per monitored parameter.

Given the noise, the model and an observation window, there is a smallest fault that is detectable at a stated confidence. It is computable rather than empirical, and it is the number a specification should contain instead of a vague claim about sensitivity.

That plateau is the interesting feature, and the parent page already explained it without knowing it had. The steady state gain of this system is one over the conductance to ambient, with the coupling conductance cancelling out entirely. Once the thermal transient has passed, the measurement contains no further information about the coupling, so every additional hour of watching adds noise and no signal.

Everything so far treated the healthy model as exact. It is not. Every unit leaves the line with capacitances, conductances and loss coefficients scattered around nominal, and a residual computed with nominal parameters is non-zero on a perfectly healthy unit simply because that unit is not nominal.

This is not a small correction. With plus or minus ten percent on the capacitances and twelve on the conductances, the healthy fleet spread swamps the sensor noise by two orders of magnitude.

The conclusion reorders the project plan. A better sensor buys nothing here. Faster sampling buys nothing. The binding constraint is manufacturing spread, and every hour spent on signal processing is an hour spent on the wrong problem.

The fix follows directly from the diagnosis. If the problem is that units differ from each other, stop comparing a unit to the fleet and start comparing it to itself.

Learn each unit's own residual over its first hours of healthy service, store that as its baseline, and monitor the departure from it. The build tolerance cancels, because it is common to the baseline and the current reading. What remains is sensor noise and genuine change.

Nine times the sensitivity for the cost of storing one number per vehicle. That is the best return available anywhere in this subject, and it is available only because the analysis identified the right limiting factor first.

The parent page built a signature matrix over four process faults and concluded that the coupling conductance was uniquely isolable. That conclusion is correct within the fault set considered, and the fault set was incomplete.

A sensor is a component and it fails. Add a two kelvin bias on each of the two temperature sensors as candidate faults, recompute the signatures, and something uncomfortable appears.

A drifting core temperature sensor and a degrading thermal interface produce residuals in exactly the same direction, minus forty five degrees, differing only in size. The monitor that was going to report a confident single cause will report the interface, a technician will replace a good thermal pad, and the sensor will keep drifting.

The signature matrix rests on an assumption that is almost never stated: that exactly one thing is wrong. Drop it and the logic changes character completely.

Two residuals span a two dimensional space. Any two independent fault directions span that same space. So for any single fault, some weighted pair of two other faults reproduces its direction exactly, and the residuals cannot tell the difference.

This is not a flaw in the method, it is arithmetic. With m relations you can distinguish at most m independent fault directions, and the number of possible multi-fault combinations grows much faster than m. Two relations and six candidate faults is a badly underdetermined problem the moment you allow more than one to be present.

• More relations. The direct fix. A third sensor gives a third dimension and the aliasing shrinks. Expensive, and worth pricing against the cost of a wrong diagnosis.
• A stated prior. Single faults really are more likely than pairs, and a Bayesian formulation can say so with a number instead of an assumption. The prior then appears in the report, where it can be argued with.
• Time. Two independent degradations rarely start on the same day. A monitor that tracks the residual direction over months will see it rotate as the second fault arrives, and a rotating direction is itself a signature of multiple faults.

Putting Sections 9 through 12 together gives a hierarchy of claims, in decreasing order of how defensible they are. A diagnostic report should say which level it is making.

The first row deserves more respect than it gets. "This unit has departed from its own baseline by six sigma in a direction consistent with the cooling path" is an enormously useful thing to put in front of a technician, and it is honest. "Replace the thermal interface material" is more satisfying to write and, on the analysis above, frequently wrong.

The parent page ended with a life distribution: a nominal of about 2,700 hours with a standard deviation of roughly 1,550. What it did not do is turn that into an action, and the step from one to the other is where most prognostics programmes quietly stop.

The mistake to avoid is scheduling at the mean. By construction, half the population fails before the mean of its own life distribution. A service interval set there converts a prognostic system into a slightly better guess about which half.

Which brings the decision down to one number: how much more an unplanned failure costs than a planned replacement. Roadside recovery, a damaged neighbouring component, a warranty claim and a customer who tells people about it, against a part and an hour of labour scheduled at a convenient time.

With that ratio in hand the optimal interval is the one minimizing expected cost per hour of service, and it can be read off directly from the life distribution.

Two things fall out that are worth carrying into a programme review. The optimal interval is never the mean and is usually a third to a half of it, so a prognostic that reports a mean and stops has done the arithmetic and withheld the answer. And narrowing the distribution is worth real money: the tighter the spread, the later you can safely schedule, which is precisely the payoff the uncertainty ranking of the parent page tells you how to buy.

Almost everything above depends on data that has to be recorded before anyone knows it will be needed. The list is short and the cost of getting it wrong is measured in vehicle-years.

• The per-unit baseline, with its validity record. When it was learned, under what conditions, and every event that invalidated it. Without this, Section 10's factor of nine is unavailable forever.
• Control effort, adaptation values and saturation flags, not just tracking errors. The parent page's masking result says the loop hides the fault in exactly the signal most people do not log.
• The residual itself, at low rate, not only the alarm bit. An alarm tells you the threshold was crossed. The residual history tells you when it started, how fast it moved, and whether its direction rotated, which is the multiple-fault evidence from Section 12.
• The duty cycle in bins, not as an average. Damage laws are exponential or power laws, so an average understates the damage by a factor of three in the parent page's own example. Bin the temperature and the load and accumulate the damage integral on the vehicle.
• The model version and the parameter set used. A residual is meaningless without knowing which model produced it, and calibrations change over a programme.

1. Derive the relation symbolically, and check detectability and isolability before anything else. That is the parent page.
2. Propagate the sensor noise through the relation in closed form and compare it to the expected fault signal. If the ratio is below one, the statistic is wrong, not the threshold.
3. Compute the ceiling: how much the fault changes the measured signal, against the noise. That is what any detector can achieve at best.
4. Choose the statistic to approach that ceiling. Matched filter if the waveform is known, parameter estimation if the size is not, a filtered residual only if the relation has low order.
5. Add the build tolerance and recompute the smallest detectable fault. If tolerance dominates, stop working on signal processing and implement a per-unit baseline.
6. Set the threshold from a stated false alarm budget over the whole service life and the whole fleet, not from three sigma.
7. Use a cumulative test for anything that degrades slowly, which is everything that wears.
8. Put sensor faults in the candidate set and recheck isolability. Break collinear ties with plausibility tests outside the residual space.
9. State the multiple-fault assumption explicitly, and pick the claim level from Section 13 that the evidence actually supports.
10. Convert the life distribution into an interval using the cost ratio, and report the interval rather than the mean.
11. Log the baseline, the residual history, the duty bins and the model version.

Noise gain

How much a residual amplifies measurement noise. Computable in closed form from the relation, and the first thing to check.

Matched filter

Correlating the data against the waveform the fault would produce. Optimal for a known signal in white noise, and the ceiling every other detector is measured against.

Separation index (d-prime)

Distance between the healthy and faulted distributions of a statistic, in units of their pooled spread. A single number for how detectable something is.

False alarm budget

The rate of wrong alarms you are willing to accept over the whole service life and the whole fleet. Thresholds should be derived from it, not the reverse.

CUSUM

A sequential test that accumulates small persistent deviations. Detects slow drifts far earlier than a fixed limit at equal false alarm rate.

Minimum detectable fault

The smallest parameter change detectable at a stated confidence, given the noise and the window. The number a specification should contain.

Per-unit baseline

Each unit's own healthy residual, learned in service and subtracted thereafter, so build tolerance cancels.

Collinear faults

Two faults producing residuals in the same direction. Never separable by any test on those residuals, whatever their magnitudes.

Signature alias

A combination of several faults reproducing the signature of a single one. Unavoidable when candidate faults outnumber relations.

Cost ratio

Cost of an unplanned failure divided by the cost of a planned replacement. The one number that converts a life distribution into a service interval.

1. Mazouchi, M. (2026). Letters Before Numbers. The parent monograph. Parts V and X in particular are assumed here.
2. Gertler, J. (1998). Fault Detection and Diagnosis in Engineering Systems. Marcel Dekker. Structured and directional residuals, and the geometry of Section 11.
3. Blanke, M., Kinnaert, M., Lunze, J., and Staroswiecki, M. (2016). Diagnosis and Fault-Tolerant Control, 3rd edition. Springer.
4. Isermann, R. (2005). Model-based fault-detection and diagnosis: status and applications. Annual Reviews in Control, 29(1), 71-85. sciencedirect.com
5. Basseville, M. and Nikiforov, I. V. (1993). Detection of Abrupt Changes: Theory and Application. Prentice Hall. The standard treatment of CUSUM and sequential detection. people.irisa.fr/Michele.Basseville/kniga
6. Page, E. S. (1954). Continuous inspection schemes. Biometrika, 41(1-2), 100-115. The original cumulative sum test.
7. Van Trees, H. L. (2001). Detection, Estimation, and Modulation Theory, Part I. Wiley. Matched filtering and the invariance argument of Section 3.
8. Kay, S. M. (1998). Fundamentals of Statistical Signal Processing, Volume II: Detection Theory. Prentice Hall.
9. Chen, J. and Patton, R. J. (1999). Robust Model-Based Fault Diagnosis for Dynamic Systems. Kluwer. Disturbance decoupling and robustness to model uncertainty.
10. Frank, P. M. and Ding, X. (1997). Survey of robust residual generation and evaluation methods in observer-based fault detection systems. Journal of Process Control, 7(6), 403-424.
11. Sankararaman, S. and Goebel, K. (2015). Uncertainty in prognostics and systems health management. International Journal of Prognostics and Health Management, 6(4). papers.phmsociety.org
12. Jardine, A. K. S., Lin, D., and Banjevic, D. (2006). A review on machinery diagnostics and prognostics implementing condition-based maintenance. Mechanical Systems and Signal Processing, 20(7), 1483-1510. The maintenance decision framing of Part IV.
13. Barlow, R. E. and Proschan, F. (1965). Mathematical Theory of Reliability. Wiley. Age replacement policies and the cost-rate minimization of Section 15.
14. Goldberg, D. (1991). What every computer scientist should know about floating-point arithmetic. ACM Computing Surveys, 23(1), 5-48. dl.acm.org